The propagation constant is a measure of change in amplitude and phase per unit distance of a sinusoidal signal or electromagnetic wave as it propagates through the medium. It describes the attenuation and phase shift of the signal as it propagates through the medium. The medium could be a transmission line or free space. The propagation constant, represented by the symbol γ (gamma), is a complex number with real and imaginary parts. It is an important property of the transmission line and is a dimensionless quantity. This constant is used in the study of wave propagation, particularly in fields like telecommunications, signal processing, and electromagnetics.

Understanding the propagation constant is crucial for designing and optimizing communication systems, designing filter and high-frequency electronic circuits, and two-port networks in signal processing. Let’s understand this constant.

The propagation constant (γ) is expressed as below:

γ = α + j β 

Here, 

α – is the real part, represents the attenuation constant, unit is dB/Km. 

The attenuation constant (α) shows the signal or electromagnetic wave attenuation (dB) per meter or km when propagating via a medium (e.g., transmission line). The signal attenuation (dB) in the transmission line is due to conductor resistive losses (including skin effect) and dielectric loss. In short, the attenuation constant shows how much the attenuation (i.e., reduction in signal amplitude) occurs per meter or Km when the signal or wave propagates through the medium. Its unit is dB/Km.

β – is the imaginary part, represents the phase constant, unit is rad/Km.

The phase constant (β) shows the signal or electromagnetic wave phase variation per meter or km when propagating via a medium (e.g., transmission line). 

A transmission line has four primary transmission line parameters: resistance (R), inductance (L), capacitance (L), and shunt conductance (G). By using the following formula, the propagation constant (γ) can be calculated from the primary transmission line parameters (R-L-C-G).

Here, R – represents the resistance of the transmission line conductors in ohms per unit length [Ω/m];

 L-    Represents the inductance of conductors in Henries per unit length [H/m];

C – Represents the capacitance between the conductors in Farad per unit length [F/m];

G – Represents the conductance (reciprocal of resistance) of dielectric material that separates the two conductors, measured in Siemens per unit length [S/m].  

G – Represents the conductance (reciprocal of resistance) of dielectric material that separates the two conductors, measured in Siemens per unit length [S/m].  

Here, R and G represent resistive and dielectric losses, and L and C represent the energy story in the transmission line. 

For a lossless transmission line (i.e., R = G = 0, and α = 0), real and imaginary parts of the propagation constant (γ) are given below:

The primary transmission line parameters (R-L-C-G) are also called primary line coefficients or primary line constants. Figure shows a distributed transmission line model for a two-conductor transmission line; the R-L-C-G elements are distributed uniformly along the line. 

Figure: Distributed parameter model of a two-conductor transmission line

Propagation constant calculation for cascaded networks: 

Figure: Three networks are cascaded, the propagation constant 

γ_total = γ₁+ γ₂+ γ₃

In cascaded topology, the propagation constant of each section may be simply added to find out the total propagation constant. For example, consider the “n” cascaded sections all having matching impedances, then the total propagation constant is given by:

                                                         γ_total = γ₁+ γ₂+ γ₃+….+ γ_n

Note: The attenuation constant (α) and phase constant (β) components of the propagation constant have a simple relationship to the signal wavelength (λ) and signal/wave phase velocity (Vp). These relationships are provided below.